P-wave slowness surface approximation for tilted orthorhombic media

Geophysics ◽  
2016 ◽  
Vol 81 (3) ◽  
pp. C99-C112 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas
Keyword(s):  
Phase Shift ◽  
Ray Tracing ◽  
Approximate Formula ◽  
P Wave ◽  
Slowness Surface ◽  
P Waves ◽  
Time Approximation ◽  
Numerical Examples ◽  
Validity Range ◽  
Plane P Waves

We have developed an analytic and approximate formula for vertical slowness components of down- and upgoing plane P waves in 3D tilted orthorhombic media. A perturbation method and Shanks transform were used to derive the approximation for slowness surface of P waves in tilted orthorhombic media. We have also quantitatively described the validity range of the radial horizontal slowness components for the proposed formula. The validity range was affected by the strength of the anellipticity of an orthorhombic medium: the stronger the anellipticity, the smaller the validity range. Numerical examples determined that the proposed formula is accurate for tilted orthorhombic media with weak to strong anellipticity. We have also evaluated in detail the application of the proposed formula on calculating the P-wave intercept time in the [Formula: see text] domain for horizontally layered, tilted orthorhombic models. Our formula is useful for ray tracing, phase-shift migration, and [Formula: see text] domain intercept time approximation for tilted orthorhombic media.

scholarly journals Phased array compaction cell for measurement of the transversely isotropic elastic properties of compacting sediments

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA113-WA123 ◽  
Author(s):  
Kurt T. Nihei ◽  
Seiji Nakagawa ◽  
Frederic Reverdy ◽  
Larry R. Myer ◽  
Luca Duranti ◽  
...  
Keyword(s):  
Elastic Properties ◽  
Phased Array ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
P Waves ◽  
Array Measurements ◽  
Experimental Apparatus ◽  
Marine Sediment Sample ◽  
Plane P Waves

Sediments undergoing compaction typically exhibit transversely isotropic (TI) elastic properties. We present a new experimental apparatus, the phased array compaction cell, for measuring the TI elastic properties of clay-rich sediments during compaction. This apparatus uses matched sets of P- and S-wave ultrasonic transducers located along the sides of the sample and an ultrasonic P-wave phased array source, together with a miniature P-wave receiver on the top and bottom ends of the sample. The phased array measurements are used to form plane P-waves that provide estimates of the phase velocities over a range of angles. From these measurements, the five TI elastic constants can be recovered as the sediment is compacted, without the need for sample unloading, recoring, or reorienting. This paper provides descriptions of the apparatus, the data processing, and an application demonstrating recovery of the evolving TI properties of a compacting marine sediment sample.

SCATTERING OF COMPRESSION WAVES BY SPHERICAL OBSTACLES

Geophysics ◽  
1959 ◽  
Vol 24 (1) ◽  
pp. 30-39 ◽  
Author(s):  
Leon Knopoff
Keyword(s):  
Wave Beam ◽  
Wave Length ◽  
Incident Beam ◽  
P Wave ◽  
Rigid Sphere ◽  
S Wave ◽  
P Waves ◽  
Compression Waves ◽  
Plane P Waves ◽  
Special Case

The scattering of plane P waves by a spherical obstacle is formulated. A calculation is given for the special case of scattering by a perfectly rigid sphere in which the medium outside has a Poisson’s ratio of [Formula: see text]. The range of sizes of obstacles used in the calculation includes radii very small compared with wave length and radii comparable to the wave length. For incident P waves, scattered P and S are computed with shifts in time phase occurring in both with respect to the incident beam. For small obstacles, the scattered S wave is generally broadside to the scattered P‐wave beam.

Geometrical spreading in a layered transversely isotropic medium with vertical symmetry axis

Geophysics ◽  
2003 ◽  
Vol 68 (6) ◽  
pp. 2082-2091 ◽  
Author(s):  
Bjørn Ursin ◽  
Ketil Hokstad
Keyword(s):  
Ray Tracing ◽  
Seismic Data ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Analytical Approximation ◽  
Geometrical Spreading ◽  
S Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
Amplitude Versus

Compensation for geometrical spreading is important in prestack Kirchhoff migration and in amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of seismic data. We present equations for the relative geometrical spreading of reflected and transmitted P‐ and S‐wave in horizontally layered transversely isotropic media with vertical symmetry axis (VTI). We show that relatively simple expressions are obtained when the geometrical spreading is expressed in terms of group velocities. In weakly anisotropic media, we obtain simple expressions also in terms of phase velocities. Also, we derive analytical equations for geometrical spreading based on the nonhyperbolic traveltime formula of Tsvankin and Thomsen, such that the geometrical spreading can be expressed in terms of the parameters used in time processing of seismic data. Comparison with numerical ray tracing demonstrates that the weak anisotropy approximation to geometrical spreading is accurate for P‐waves. It is less accurate for SV‐waves, but has qualitatively the correct form. For P waves, the nonhyperbolic equation for geometrical spreading compares favorably with ray‐tracing results for offset‐depth ratios less than five. For SV‐waves, the analytical approximation is accurate only at small offsets, and breaks down at offset‐depth ratios less than unity. The numerical results are in agreement with the range of validity for the nonhyperbolic traveltime equations.

Simulation of thermoelastic waves based on the Lord-Shulman theory

Geophysics ◽  
2021 ◽  
Vol 86 (3) ◽  
pp. T155-T164
Author(s):  
Wanting Hou ◽  
Li-Yun Fu ◽  
José M. Carcione ◽  
Zhiwei Wang ◽  
Jia Wei
Keyword(s):  
Thermal Conductivity ◽  
Expansion Coefficient ◽  
Velocity Dispersion ◽  
Wave Attenuation ◽  
Splitting Method ◽  
Grazing Incidence ◽  
P Wave ◽  
Staggered Grid ◽  
Thermal Mode ◽  
P Waves

Thermoelasticity is important in seismic propagation due to the effects related to wave attenuation and velocity dispersion. We have applied a novel finite-difference (FD) solver of the Lord-Shulman thermoelasticity equations to compute synthetic seismograms that include the effects of the thermal properties (expansion coefficient, thermal conductivity, and specific heat) compared with the classic forward-modeling codes. We use a time splitting method because the presence of a slow quasistatic mode (the thermal mode) makes the differential equations stiff and unstable for explicit time-stepping methods. The spatial derivatives are computed with a rotated staggered-grid FD method, and an unsplit convolutional perfectly matched layer is used to absorb the waves at the boundaries, with an optimal performance at the grazing incidence. The stability condition of the modeling algorithm is examined. The numerical experiments illustrate the effects of the thermoelasticity properties on the attenuation of the fast P-wave (or E-wave) and the slow thermal P-wave (or T-wave). These propagation modes have characteristics similar to the fast and slow P-waves of poroelasticity, respectively. The thermal expansion coefficient has a significant effect on the velocity dispersion and attenuation of the elastic waves, and the thermal conductivity affects the relaxation time of the thermal diffusion process, with the T mode becoming wave-like at high thermal conductivities and high frequencies.

Estimates of velocity structure and source depth using multiple P waves from aftershocks of the 1987 Elmore Ranch and Superstition Hills, California, earthquakes

1991 ◽  
Vol 81 (2) ◽  
pp. 508-523
Author(s):  
Jim Mori
Keyword(s):  
Velocity Structure ◽  
Velocity Model ◽  
P Wave ◽  
Sediment Layer ◽  
Source Depth ◽  
Imperial Valley ◽  
P Waves ◽  
Main Shocks ◽  
Aftershock Sequences ◽  
Event Record

Abstract Event record sections, which are constructed by plotting seismograms from many closely spaced earthquakes recorded on a few stations, show multiple free-surface reflections (PP, PPP, PPPP) of the P wave in the Imperial Valley, California. The relative timing of these arrivals is used to estimate the strength of the P-wave velocity gradient within the upper 5 km of the sediment layer. Consistent with previous studies, a velocity model with a value of 1.8 km/sec at the surface increasing linearly to 5.8 km/sec at a depth of 5.5 km fits the data well. The relative amplitudes of the P and PP arrivals are used to estimate the source depth for the aftershock distributions of the Elmore Ranch and Superstition Hills main shocks. Although the depth determination has large uncertainties, both the Elmore Ranch and Superstition Hills aftershock sequences appear to have similar depth distribution in the range of 4 to 10 km.

Reflection moveout approximations for P-waves in a moderately anisotropic homogeneous tilted transverse isotropy layer

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. C175-C185 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
3D Space ◽  
Axis Of Symmetry ◽  
Relative Errors ◽  
Symmetry Tests ◽  
Ti Media

We have developed approximate nonhyperbolic P-wave moveout formulas applicable to weakly or moderately anisotropic media of arbitrary anisotropy symmetry and orientation. Instead of the commonly used Taylor expansion of the square of the reflection traveltime in terms of the square of the offset, we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. No acoustic approximation is used. We specify the formulas designed for anisotropy of arbitrary symmetry for the transversely isotropic (TI) media with the axis of symmetry oriented arbitrarily in the 3D space. Resulting formulas depend on three P-wave WA parameters specifying the TI symmetry and two angles specifying the orientation of the axis of symmetry. Tests of the accuracy of the more accurate of the approximate formulas indicate that maximum relative errors do not exceed 0.3% or 2.5% for weak or moderate P-wave anisotropy, respectively.

scholarly journals Physiological ECG Value for Polish Half-Bred Anglo-Arab Horses

2012 ◽  
Vol 56 (4) ◽  
pp. 631-635 ◽  
Keyword(s):  
Heart Rate ◽  
P Wave ◽  
P Waves ◽  
T Wave ◽  
Duration Time ◽  
Qt Intervals ◽  
Electrocardiographic Examination ◽  
Electrocardiographic Findings ◽  
Almost All ◽  
T Waves

Abstract The electrocardiographic examination was performed in 33 training horses (2-16 years of age, 11 males and 22 females). Einthoven and precordial leads (I, II, III, aVR, aVL, aVF, CV1, CV2, CV4) were used. The ECG was performed in resting horses and immediately after exercise (10 min walk, 15 min trot, 10 min canter) using a portable Schiller AT-1 3-channel electrocardiograph, with a paper speed of 25 mm sec-1 and a sensitivity of 10 mm.mV-1. The heart rate, wave amplitudes, and duration time were estimated manually. All horses presented a significant increase in heart rate after exercise (rest 43.83 ±10.33 vs. exercise 73.2 ±14.8). QT intervals were significantly shortened in most of the leads. In resting horses, all P waves in the lead I were positive and almost all II, III and CV4 leads were positive. Simple negative P wave dominated in aVR and only simple negative T wave was found in the leads I. The biphasic shape was observed. After exercise, the amplitude of P and T waves rose, however, clear changes were not observed in wave polarisation and form. In the absence of specific racial characteristics of the electrocardiogram in the Polish Anglo- Arabians, electrocardiographic findings can be interpreted according to ECG standards adopted for horses.

THE ANOMALOUS PROCESS γπ → ππ AND ITS IMPACT ON THE π0 TRANSITION FORM FACTOR

2014 ◽  
Vol 35 ◽  
pp. 1460397
Author(s):  
BASTIAN KUBIS
Keyword(s):  
Form Factor ◽  
Phase Shift ◽  
Theoretical Analysis ◽  
Cross Section ◽  
Transition Form Factor ◽  
P Wave ◽  
Chiral Anomaly ◽  
Transition Form ◽  
Dispersive Representation ◽  
Important Input

The process γπ → ππ, in the limit of vanishing photon and pion energies, is determined by the chiral anomaly. This reaction can be investigated experimentally using Primakoff reactions, as currently done at COMPASS. We derive a dispersive representation that allows one to extract the chiral anomaly from cross-section measurements up to 1 GeV, where effects of the ρ resonance are included model-independently via the ππ P-wave phase shift. We discuss how this amplitude serves as an important input to a dispersion-theoretical analysis of the π0 transition form factor, which in turn is a vital ingredient to the hadronic light-by-light contribution to the anomalous magnetic moment of the muon.

Amplitude, Fresnel zone, and NMO velocity for PP and SS normal-incidence reflections

Geophysics ◽  
2006 ◽  
Vol 71 (2) ◽  
pp. W1-W14 ◽  
Author(s):  
Einar Iversen
Keyword(s):  
Phase Shift ◽  
Ray Tracing ◽  
Explicit Knowledge ◽  
Fresnel Zone ◽  
Normal Wave ◽  
Normal Incidence ◽  
Geometric Spreading ◽  
Dynamic Ray Tracing ◽  
The One ◽  
Paraxial Ray

Inspired by recent ray-theoretical developments, the theory of normal-incidence rays is generalized to accommodate P- and S-waves in layered isotropic and anisotropic media. The calculation of the three main factors contributing to the two-way amplitude — i.e., geometric spreading, phase shift from caustics, and accumulated reflection/transmission coefficients — is formulated as a recursive process in the upward direction of the normal-incidence rays. This step-by-step approach makes it possible to implement zero-offset amplitude modeling as an efficient one-way wavefront construction process. For the purpose of upward dynamic ray tracing, the one-way eigensolution matrix is introduced, having as minors the paraxial ray-tracing matrices for the wavefronts of two hypothetical waves, referred to by Hubral as the normal-incidence point (NIP) wave and the normal wave. Dynamic ray tracing expressed in terms of the one-way eigensolution matrix has two advantages: The formulas for geometric spreading, phase shift from caustics, and Fresnel zone matrix become particularly simple, and the amplitude and Fresnel zone matrix can be calculated without explicit knowledge of the interface curvatures at the point of normal-incidence reflection.

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